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## G = C52⋊D4order 416 = 25·13

### 3rd semidirect product of C52 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — C52⋊D4
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C2×D52 — C52⋊D4
 Lower central C13 — C2×C26 — C52⋊D4
 Upper central C1 — C22 — C2×D4

Generators and relations for C52⋊D4
G = < a,b,c | a52=b4=c2=1, bab-1=a25, cac=a-1, cbc=b-1 >

Subgroups: 840 in 108 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C13, C42, C2×D4, C2×D4, D13, C26, C26, C26, C41D4, Dic13, C52, D26, C2×C26, C2×C26, D52, C2×Dic13, C13⋊D4, C2×C52, D4×C13, C22×D13, C22×C26, C4×Dic13, C2×D52, C2×C13⋊D4, D4×C26, C52⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C41D4, D26, C13⋊D4, C22×D13, D4×D13, C2×C13⋊D4, C52⋊D4

Smallest permutation representation of C52⋊D4
On 208 points
Generators in S208
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52)(53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208)
(1 98 192 156)(2 71 193 129)(3 96 194 154)(4 69 195 127)(5 94 196 152)(6 67 197 125)(7 92 198 150)(8 65 199 123)(9 90 200 148)(10 63 201 121)(11 88 202 146)(12 61 203 119)(13 86 204 144)(14 59 205 117)(15 84 206 142)(16 57 207 115)(17 82 208 140)(18 55 157 113)(19 80 158 138)(20 53 159 111)(21 78 160 136)(22 103 161 109)(23 76 162 134)(24 101 163 107)(25 74 164 132)(26 99 165 105)(27 72 166 130)(28 97 167 155)(29 70 168 128)(30 95 169 153)(31 68 170 126)(32 93 171 151)(33 66 172 124)(34 91 173 149)(35 64 174 122)(36 89 175 147)(37 62 176 120)(38 87 177 145)(39 60 178 118)(40 85 179 143)(41 58 180 116)(42 83 181 141)(43 56 182 114)(44 81 183 139)(45 54 184 112)(46 79 185 137)(47 104 186 110)(48 77 187 135)(49 102 188 108)(50 75 189 133)(51 100 190 106)(52 73 191 131)
(1 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 136)(54 135)(55 134)(56 133)(57 132)(58 131)(59 130)(60 129)(61 128)(62 127)(63 126)(64 125)(65 124)(66 123)(67 122)(68 121)(69 120)(70 119)(71 118)(72 117)(73 116)(74 115)(75 114)(76 113)(77 112)(78 111)(79 110)(80 109)(81 108)(82 107)(83 106)(84 105)(85 156)(86 155)(87 154)(88 153)(89 152)(90 151)(91 150)(92 149)(93 148)(94 147)(95 146)(96 145)(97 144)(98 143)(99 142)(100 141)(101 140)(102 139)(103 138)(104 137)(157 162)(158 161)(159 160)(163 208)(164 207)(165 206)(166 205)(167 204)(168 203)(169 202)(170 201)(171 200)(172 199)(173 198)(174 197)(175 196)(176 195)(177 194)(178 193)(179 192)(180 191)(181 190)(182 189)(183 188)(184 187)(185 186)```

`G:=sub<Sym(208)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,98,192,156)(2,71,193,129)(3,96,194,154)(4,69,195,127)(5,94,196,152)(6,67,197,125)(7,92,198,150)(8,65,199,123)(9,90,200,148)(10,63,201,121)(11,88,202,146)(12,61,203,119)(13,86,204,144)(14,59,205,117)(15,84,206,142)(16,57,207,115)(17,82,208,140)(18,55,157,113)(19,80,158,138)(20,53,159,111)(21,78,160,136)(22,103,161,109)(23,76,162,134)(24,101,163,107)(25,74,164,132)(26,99,165,105)(27,72,166,130)(28,97,167,155)(29,70,168,128)(30,95,169,153)(31,68,170,126)(32,93,171,151)(33,66,172,124)(34,91,173,149)(35,64,174,122)(36,89,175,147)(37,62,176,120)(38,87,177,145)(39,60,178,118)(40,85,179,143)(41,58,180,116)(42,83,181,141)(43,56,182,114)(44,81,183,139)(45,54,184,112)(46,79,185,137)(47,104,186,110)(48,77,187,135)(49,102,188,108)(50,75,189,133)(51,100,190,106)(52,73,191,131), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,136)(54,135)(55,134)(56,133)(57,132)(58,131)(59,130)(60,129)(61,128)(62,127)(63,126)(64,125)(65,124)(66,123)(67,122)(68,121)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,105)(85,156)(86,155)(87,154)(88,153)(89,152)(90,151)(91,150)(92,149)(93,148)(94,147)(95,146)(96,145)(97,144)(98,143)(99,142)(100,141)(101,140)(102,139)(103,138)(104,137)(157,162)(158,161)(159,160)(163,208)(164,207)(165,206)(166,205)(167,204)(168,203)(169,202)(170,201)(171,200)(172,199)(173,198)(174,197)(175,196)(176,195)(177,194)(178,193)(179,192)(180,191)(181,190)(182,189)(183,188)(184,187)(185,186)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,98,192,156)(2,71,193,129)(3,96,194,154)(4,69,195,127)(5,94,196,152)(6,67,197,125)(7,92,198,150)(8,65,199,123)(9,90,200,148)(10,63,201,121)(11,88,202,146)(12,61,203,119)(13,86,204,144)(14,59,205,117)(15,84,206,142)(16,57,207,115)(17,82,208,140)(18,55,157,113)(19,80,158,138)(20,53,159,111)(21,78,160,136)(22,103,161,109)(23,76,162,134)(24,101,163,107)(25,74,164,132)(26,99,165,105)(27,72,166,130)(28,97,167,155)(29,70,168,128)(30,95,169,153)(31,68,170,126)(32,93,171,151)(33,66,172,124)(34,91,173,149)(35,64,174,122)(36,89,175,147)(37,62,176,120)(38,87,177,145)(39,60,178,118)(40,85,179,143)(41,58,180,116)(42,83,181,141)(43,56,182,114)(44,81,183,139)(45,54,184,112)(46,79,185,137)(47,104,186,110)(48,77,187,135)(49,102,188,108)(50,75,189,133)(51,100,190,106)(52,73,191,131), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,136)(54,135)(55,134)(56,133)(57,132)(58,131)(59,130)(60,129)(61,128)(62,127)(63,126)(64,125)(65,124)(66,123)(67,122)(68,121)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,105)(85,156)(86,155)(87,154)(88,153)(89,152)(90,151)(91,150)(92,149)(93,148)(94,147)(95,146)(96,145)(97,144)(98,143)(99,142)(100,141)(101,140)(102,139)(103,138)(104,137)(157,162)(158,161)(159,160)(163,208)(164,207)(165,206)(166,205)(167,204)(168,203)(169,202)(170,201)(171,200)(172,199)(173,198)(174,197)(175,196)(176,195)(177,194)(178,193)(179,192)(180,191)(181,190)(182,189)(183,188)(184,187)(185,186) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)], [(1,98,192,156),(2,71,193,129),(3,96,194,154),(4,69,195,127),(5,94,196,152),(6,67,197,125),(7,92,198,150),(8,65,199,123),(9,90,200,148),(10,63,201,121),(11,88,202,146),(12,61,203,119),(13,86,204,144),(14,59,205,117),(15,84,206,142),(16,57,207,115),(17,82,208,140),(18,55,157,113),(19,80,158,138),(20,53,159,111),(21,78,160,136),(22,103,161,109),(23,76,162,134),(24,101,163,107),(25,74,164,132),(26,99,165,105),(27,72,166,130),(28,97,167,155),(29,70,168,128),(30,95,169,153),(31,68,170,126),(32,93,171,151),(33,66,172,124),(34,91,173,149),(35,64,174,122),(36,89,175,147),(37,62,176,120),(38,87,177,145),(39,60,178,118),(40,85,179,143),(41,58,180,116),(42,83,181,141),(43,56,182,114),(44,81,183,139),(45,54,184,112),(46,79,185,137),(47,104,186,110),(48,77,187,135),(49,102,188,108),(50,75,189,133),(51,100,190,106),(52,73,191,131)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(53,136),(54,135),(55,134),(56,133),(57,132),(58,131),(59,130),(60,129),(61,128),(62,127),(63,126),(64,125),(65,124),(66,123),(67,122),(68,121),(69,120),(70,119),(71,118),(72,117),(73,116),(74,115),(75,114),(76,113),(77,112),(78,111),(79,110),(80,109),(81,108),(82,107),(83,106),(84,105),(85,156),(86,155),(87,154),(88,153),(89,152),(90,151),(91,150),(92,149),(93,148),(94,147),(95,146),(96,145),(97,144),(98,143),(99,142),(100,141),(101,140),(102,139),(103,138),(104,137),(157,162),(158,161),(159,160),(163,208),(164,207),(165,206),(166,205),(167,204),(168,203),(169,202),(170,201),(171,200),(172,199),(173,198),(174,197),(175,196),(176,195),(177,194),(178,193),(179,192),(180,191),(181,190),(182,189),(183,188),(184,187),(185,186)]])`

74 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 13A ··· 13F 26A ··· 26R 26S ··· 26AP 52A ··· 52L order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 size 1 1 1 1 4 4 52 52 2 2 26 26 26 26 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D13 D26 D26 C13⋊D4 D4×D13 kernel C52⋊D4 C4×Dic13 C2×D52 C2×C13⋊D4 D4×C26 Dic13 C52 C2×D4 C2×C4 C23 C4 C2 # reps 1 1 1 4 1 4 2 6 6 12 24 12

Matrix representation of C52⋊D4 in GL4(𝔽53) generated by

 34 12 0 0 15 10 0 0 0 0 50 13 0 0 40 3
,
 6 9 0 0 43 47 0 0 0 0 1 0 0 0 0 1
,
 15 2 0 0 47 38 0 0 0 0 50 13 0 0 32 3
`G:=sub<GL(4,GF(53))| [34,15,0,0,12,10,0,0,0,0,50,40,0,0,13,3],[6,43,0,0,9,47,0,0,0,0,1,0,0,0,0,1],[15,47,0,0,2,38,0,0,0,0,50,32,0,0,13,3] >;`

C52⋊D4 in GAP, Magma, Sage, TeX

`C_{52}\rtimes D_4`
`% in TeX`

`G:=Group("C52:D4");`
`// GroupNames label`

`G:=SmallGroup(416,161);`
`// by ID`

`G=gap.SmallGroup(416,161);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,103,218,188,13829]);`
`// Polycyclic`

`G:=Group<a,b,c|a^52=b^4=c^2=1,b*a*b^-1=a^25,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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